If `alpha` and `beta` are roots of the equation `x^(2)-3x+1=0` and `a_(n)=alpha^(n)+beta^(n)-1` then find the value of `(a_(5)-a_(1))/(a_(3)-a_(1))`

Answer

8

Explanation

`alpha+beta=3, alphabeta=1`
`because alpha^(2)beta^(2)=7becausea_(1)=2,a_(2)=6`
Also `alpha^(n)-3alpha^(n-1)+alpha^(n-2)=0`
and `beta^(n)-3beta^(n-1)+beta^(n-2)=0`
`because(alpha^(n)+beta^(n))-3(alpha^(n-1)+beta^(n-1))+(alpha^(n-2)+beta^(n-2))=0`
`because(alpha^(n)+beta^(n)-1)-3(alpha^(n-1)+beta^(n-1)-1+(alpha^(n-2)+beta^(n-2)-1)=1`
`becausea_(n)=1+3a_(n-1)-1_(n-2)`
`becausea_(3)=17,a_(4)=46,a_(5)=122,because(a_(5)-a_(1))/(a_(3)-a_(1))=8`